Jean-Pierre Bourguignon A MATHEMATICIAN'S VISIT TO KALUZA ...

THEOREM 8 (L. BÉRARD BERGERY) - Let p : (M,g) -• (M,g) be a Riemanniansubmersion with totalìy geodesie fibres and base and fibre metrics gand g Einstein for positive constants X and À. The metric g is Einstein withConstant X ifand onìy ifi) the connection a is a Yang-MMs and its curvature Q° has Constantìength e and satisfìes157(22) ì 'f ! G/H is afibration. If we choose an Ada(#) invariant complement to the Lie algebra ofH in the Lie algebra of G, and on it an AdG(/0-invariant scalar product, weobtain a G-invariant Riemannian metric g on G/H, By proceeding similarly forthe pair (H,K), we obtain also an /Mnvariant Riemannian metric g on H/K.By taking the direct sum metric on the sum of the complements of the Liesubalgebras, we obtain a Riemannian metric g on G/K which, by a theorem ofL. Bérard-Bergery, is a Riemannian submersion with totally geodesie fibres.Many important geometrie examples fall under this category, such as theusuai Hopf fibrationsof spheres over projective spaces, and also the generalizedHopf fibrations of projective spaces over other projective spaces.Then, Theorem 8 applied to this family gives "exotic" examples of Einsteinspaces on simple manifolds. In particular, from the classical Hopf fibrations5 4,+3 —• HP*, one gets two Einstein metrics on (4g + 3)-spheres; from